Enter quadratic or square root function to automatically determine range based on vertex and opening direction
Select Function Type
Enter quadratic parameters (y = ax² + bx + c)
y =x² +x +
Enter square root parameters (y = √(ax + b), domain x ≥ -b/a)
y = √(x +)
Result
Step-by-Step Derivation
Common Function Range Rules
Quadratic y=ax²+bx+c (a>0): range [y₀, +inf), y₀=(4ac-b²)/(4a)
Quadratic y=ax²+bx+c (a<0): range (-inf, y₀], y₀=(4ac-b²)/(4a)
Square root y=√(ax+b): range [0, +inf)
Linear y=ax+b (a≠0): range (-inf, +inf)
Finding the range requires determining the function type and properties. For quadratics, use completing the square or the vertex formula. For rational functions, analyze numerator/denominator constraints.
⚠Vertex formula: x₀ = -b/(2a), extremum y₀ = (4ac-b²)/(4a). Opening upward (a>0) means y₀ is the minimum; opening downward (a<0) means y₀ is the maximum.
What Is the Range of a Function?
The range is the set of all possible output values (y) that a function can produce as x takes every value in the domain. Finding the range is a core part of function analysis.
Definition
The range is the set of all possible function outputs, forming one of the three essential elements along with domain and mapping rule.
Vertex Method
For quadratics y=ax²+bx+c, vertex x₀=-b/(2a), extremum y₀=(4ac-b²)/(4a). Opening direction determines range.
Completing the Square
Rewrite as vertex form y = a(x-x₀)² + y₀ to directly read the vertex and determine the range.
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